## Elasticity

 Unit code: MATH35021 Credit Rating: 10 Unit level: Level 3 Teaching period(s): Semester 1 Offered by School of Mathematics Available as a free choice unit?: N

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#### Aims

The course unit introduces students to the mathematical theory of linear elasticity. The general theory is developed and then applied to a number of practical problems in solid mechanics. Analytical techniques will be introduced to solve the resulting differential equations.

#### Overview

This course unit gives an introduction to the linearised theory of elasticity. A typical problem of the subject is as follows: Suppose an elastic body (e.g. an underground oil pipe) is subjected to some loading on its outer surface. What is the stress distribution which is generated throughout the body? Does this stress distribution have unexpectedly large values which might lead to failure? The subject is developed, and particular problems solved, from a mathematical standpoint.

#### Learning outcomes

On successful completion of this course unit students will be able to:

• identify, calculate and provide physical interpretations for displacements, strains, rotation tensors, stresses and tractions for given deformations of linear elastic materials,
• distinguish between rigid-body motions and strains and use the compatibility equations to establish the validity of strain fields,
• use constitutive laws to relate strain and stress fields,
• solve the Navier-Lamé equations given specific functional forms for the displacement field,
• solve two-dimensional problems in linear elasticity (plane strain or plane stress) using the Airy stress function or related approaches,
• formulate, solve and provide physical interpretations for the solutions of boundary value problems in linear elasticity.

#### Assessment methods

• Other - 20%
• Written exam - 80%

#### Assessment Further Information

• Coursework: weighting 20%
• End of semester examination: two hours weighting 80%

#### Syllabus

• Analysis of strain [6]: the infinitesimal strain tensor, derivation and interpretation; maximum normal strain; strain invariants; equations of compatibility of strain.
• Analysis of stress [2]: the traction vector and the stress tensor; maximum normal stress. Stress equations of motion and their linearisation.
• Constitutive equations [1] stress-strain relations. Elastic and linearly elastic materials; isotropic materials.
• Governing Equations [1]: Navier's equation of motion for the displacement vector; equations of compatibility of stress for an isotropic materials in equilibrium (Beltrami-Michell equations).
• Formulation of boundary value problems of linear elastostatics [12]:  One-dimensional problems. A selection of soluble problems (which are effectively one-dimensional) in Cartesian, cylindrical polar or spherical polar coordinates. St. Venant's principle.  Plane strain problems. Theory of plane strain, Airy stress function. A selection of soluble two-dimensional problems using plane-strain theory.

The course does not follow one particular book. A good book, covering most of the course, is

• P.L. Gould, Introduction to Linear Elasticity, 2nd Edition, Springer, 1994.

This book, and many others on the theory of elasticity can be found at 531.38 in the John Rylands University Library.

#### Feedback methods

Tutorials will provide an opportunity for students' work on the weekly example sheets and coursework to be discussed and
provide feedback on their understanding.  The extended coursework calculation tests understanding and also provides another opportunity for students to receive feedback.  Students can also get feedback directly from the lecturer by making an appointment, for example during the lecturer's office hours.

#### Study hours

• Lectures - 22 hours
• Tutorials - 11 hours
• Independent study hours - 67 hours

#### Teaching staff

Tom Shearer - Unit coordinator